Identification of extended Hammerstein systems with hysteresis-type input nonlinearities described by Preisach model

Abstract

This paper introduces Preisach model to describe hysteretic input nonlinearities of extended Hammerstein systems. Preisach model is proved to be an effective model to cover most of hysteretic input nonlinearities studied in the existing literatures on identification of extended Hammerstein systems. That is, Preisach model encloses the backlash-type nonlinearities and the hysteresis-relay nonlinearities as special cases, but excludes the switch-type nonlinearities. Hence, the usage of Preisach model is able to resolve an issue that types of hysteretic input nonlinearities are often inaccessible in practice, but most of existing identification methods have to know the type of input hysteretic nonlinearity a priori. For identification of extended Hammerstein systems with input nonlinearity described by Presiach model, the persistently exciting (PE) condition of input is established and the consistency of estimated model parameters is proved. In particular, the requirement on inputs to meet with the PE condition covers a wide class of signals, which resolves another issue in the existing studies that specially designed inputs are usually required. Numerical and industrial examples are provided to illustrate the obtained results.

Appendices

Appendix 1: Representation theorem

This appendix recalls the Representation Theorem established by Mayergoyz [18] that gives a sufficient and necessary condition for the hysteresis-type nonlinearities that can be represented by Preisach model:

Theorem 1

(Representation Theorem) [18] The wiping-out property and the congruency property constitute the necessary and sufficient condition for a rate-independent hysteresis nonlinearity to be represented by Preisach model on the set of piecewise continuous inputs.

The three properties in Theorem 1 are defined as follows:

Property 1

(Wiping-out property) [19] (pp. 15–17 therein) Only the series of dominant input extrema are stored by Preisach model, and all other input local extrema are wiped out.

Property 2

(Congruency property) [19] (pp. 18–19 therein) All hysteresis loops corresponding to the same extreme values of input are congruent in the geometric sense (i.e., the loops are identical up to translation).

Property 3

(Rate-independence property) [28] (p. 13 therein) The hysteresis nonlinearity output is invariant to any increasing time homeomorphism (i.e., the output is independent to the variation velocity of the input).

Appendix 2: First-order reversal function

This appendix serves to recall the first-order reversal function defined in [19].

First, the geometric interpretation of Preisach model is a useful tool [18, 19]. Define two sets $S^+(t)$, $S^-(t)$ and one memory curve $L(t)$ as follows:

• $S^+(t)$ consists of all the points $(\beta , \alpha )$ for which the relay operator $\gamma _{\beta \alpha }(t) = 1$ at the time instant $t$;

• $S^-(t)$ consists of all the points $(\beta , \alpha )$ for which $\gamma _{\beta \alpha }(t) = -1$ at the time instant $t$;

• $L(t)$ is the boundary line between $S^+(t)$ and $S^-(t)$.

With the sets $S^+(t)$ and $S^-(t)$, Preisach model in (3) can be equivalently represented as

$$\begin{aligned} f(t) = \int \int \limits _{S^+(t)} \mu (\beta , \alpha )\text{ d }\beta \text{ d }\alpha - \int \int \limits _{S^-(t)} \mu (\beta , \alpha ) \text{ d }\beta \text{ d }\alpha . \end{aligned}$$

(42)

Second, the first-order descending reversal function is formulated as the path of $f(t)$ for a monotonic increment of $u(t)$ starting from the minimum value $u_\mathrm{{min}}$ to the value $\alpha '$ followed by a subsequent monotonic decrement to $\beta '$; the corresponding $\alpha -\beta$ diagram is shown in Fig. 13. Denote $f_{\alpha '}$ and $f_{\alpha ', \beta '}$ as the output values of $f(t)$ in (42) for $u(t) = \alpha '$ and $u(t)=\beta '$, respectively. The so-called Everett function is defined as [18],

$$\begin{aligned} F(\beta ', \alpha ') = \frac{1}{2}\left( f_{\alpha '} - f_{\alpha '\beta '}\right) . \end{aligned}$$

(43)

It is revealed from the $\alpha -\beta$ diagrams in Fig. 13 that the triangle $T(\beta ',\alpha ')$ with three vertices $(\beta ', \beta ')$, $(\alpha ', \alpha ')$ and $(\beta ', \alpha ')$ is added to the negative set $S^-$ and subtracted from the positive set $S^+$, when $u(t)$ is monotonically decreased from $\alpha '$ to $\beta '$. From (42) and (43), we have

$$\begin{aligned} F(\beta ', \alpha ') = \int \int _{T(\beta ',\alpha ')} \mu (\beta , \alpha ) \mathrm {d}\beta \mathrm {d}\alpha . \end{aligned}$$

Thus, the weighting function $\mu (\beta ', \alpha ')$ can be determined by differentiating $F(\beta ', \alpha ')$ with respect to $\beta '$ and $\alpha '$, i.e.,

$$\begin{aligned} \mu (\beta ', \alpha ') = \frac{\partial ^{2}{F}(\beta ', \alpha ')}{\partial {\beta '}\partial {\alpha '}}. \end{aligned}$$

(44)

Fig. 13

The memory curve $L(t)$ and the sets $S^+(t)$ and $S^-(t)$: a the input $u(t)$ is first increased from $u_\mathrm{{min}}$ to $\alpha '$, and b $u$ is decreased to $\beta '$ afterward